Survival analysis methods in Insurance Applications in car insurance contracts

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Survival analysis methods in Insurance Applications in car insurance contracts"

Transcription

1 Survival analysis methods in Insurance Applications in car insurance contracts Abder OULIDI 1-2 Jean-Marie MARION 1 Hérvé GANACHAUD 3 1 Institut de Mathématiques Appliquées (IMA) Angers France 2 Institut de Statistiques et d Economie Appliquées (INSEA) Rabat Maroc 3 Mutuelles du Mans Assurances (MMA) Le Mans - France

2 Context Solvency II derectives to map, to identify their own risks to analyse and modelise their own risks Car insurance mature market Competition expending ( banks-insurers ) Quasi stability insurable motor vehicle populationp Insurers are led to develop optimal models of surveilance and mangement of their portfolio.

3 Plan 1. Introduction : definitions and notations 2. Survival models 2.1 Non parametic models 2.2 Parametric models 2.3 Semi parametric models 3. Application 3.1 Data set Results 4. Conclusion and perspectives

4 1- INTRODUCTION: Applied Fields Statutory Mortality Tables. Experience Mortality Tables. Insurance Contracts.

5 1- INTRODUCTION: Definitions and notations T survival time from the starting point until cancellation of a contract. f probability density function and F cumulative distribution function of the distribution of T. S(t)=P(T>t) survival function. t hazard function defined d by: f () t 1 t lim P t T tt/ T t St () t 0 t

6 1- INTRODUCTION: Definitions and notations A(t) cumulative hazard function defined by : A () t t s ds 0 St ( ) exp At ( ) since S(0) 1

7 2- SURVIVAL MODELS: Non parametric ti models Kaplan-Meier estimator Peterson estimator Nelson estimator

8 2- SURVIVAL MODELS: Parametric models ( t1,..., t n ) a possibly right and left censored set of observations from: t z ln i i i the distribution of the error term i can be specified as exponential, Weibull, log normal, log logistic distributions

9 2- SURVIVAL MODELS: Semi-parametric ti models Cox model with time-fixed covariates: t / z t exp z β a vect or of r egr essi on par a met er s z a vect or of covari at es val ues 0 0 an unspecifi ed baseli ne hazar d f uncti on

10 2- SURVIVAL MODELS: Semi-parametric ti models The Cox regression model is a proportional hazard model t / z1 the «hazard ratio» exp z is independant of t 11z2 2 t/ z 2

11 2- SURVIVAL MODELS: Semi-parametric ti models t,..., t 1 n a sample of orderly observations. In order to estimate we use the «partial likelihood function»: n expz i k krt i,..., t ; 1 n i1 exp z Lt i

12 2- SURVIVAL MODELS: Semi-parametric ti models How to test proportional hazard assumption? Plots of Log cumulative hazard rate. Scaled Schoenfeld residuals an alternative to proportional hazards is time varying coefficients t g t If 0 the «hazard ratio» is not constant with respect to time t.

13 2- SURVIVAL MODELS: Semi-parametric ti models Alternatives models: A- Cox model with time-dependant covariates: t/ z texp z t The «partial likelihood function» is defined by: exp n zi t i Lt,..., t ; 1 n i1 exp zk tk krt i 0 i

14 2- SURVIVAL MODELS B- Non parametric Aalen s additive regression model: 0 t/ Z t t t Z( t) Our data, based on a sample of size n, consist of the triple Ti, i, Zi t i the event indicator for the ith contract t

15 2- SURVIVAL MODELS Aalen s additive regression model We define: i i T t; 1 Nt ( ) N t avec N t 1 1 i n i 1 i n i T t Y () t Y t avec Y t 1 (observation at risk at t - ) i i i

16 2- SURVIVAL MODELS Aalen s additive regression model The additive hazard model can be written in matrix form: dn () t Y () t db () t dm () t Y( t) is the matrix multiplicative intensity model M ( t) is a mean zero martingale k k k B ( t) B t with B t s ds 1kp t 0

17 2- SURVIVAL MODELS Aalen s additive regression model The least square estimator for B(t) is given by: Bt YTYT YT T 1 ˆ i i i 1( i) it ; t i where 1 T is a vector with ith element equal to 1 if contract i is cancelled i An estimate of t is given by the slope of the estimate or by using smoothing techniques k Bˆk t

18 2- SURVIVAL MODELS Aalen s additive regression model The estimator of the covariance matrix of ˆB t is: it ; t Var Bˆ ( t) Y Ti Y T i Y Ti 1 Ti 1 Ti Y T i Y Ti Y T i i 1 1 The hypothesis of no regression effect for one or more covariates is testing by: ( H ) B t 0 0 k

19 Dataset t Dataset from French insurance company 1461 car s insurance contracts t created during the period of June 13th, 1974 to December 28th, Cancellation of a contract could only be observed after January 1st, If the cancelling contract is before February 7th, 2006 we have considered the duration between cancellation and conclusion of contract (otherwise right censoring).

20 Dataset Lifetime variable : lifespan of cars insurance (Durvie) If cancellation is before February, 7th, 2006 Durvie = contract cancellation s date - contract conclusion s date If cancellation is after February, 7th, 2006 Durvie = February,7th, contract conclusion s s date fixed right censoring date

21 Dataset Covariates: Age of vehicle (AgeVehic) If AgeVehic 1 AgeVehic1 If 1<AgeVehic 4 AgeVehic2 If 4<AgeVehic 8 AgeVehic3 If 8<AgeVehic AgeVehic4 Type of insurance (Formule) Tierce Intégrale (formule tous risques) Formule1 Tierce Maxi (formule RC + dommages) Formule2 Tierce Simple (formule RC seule) Formule3

22 Dataset Bonus-Malus variable (BM) If Bonus-Malus = (b bonus 50%) BM1 If 0.5 < Bonus-Malus 0.7 (30 % bonus <50 %) BM2 If Bonus-Malus > 0.7 (bonus or malus < 30 % ) BM3

23 Results All Censoring Cancellation Effectifs Number of contracts All Cens. cancel BM BM All Cens. cancel Formule Formule AgeVehic 1 AgeVehic 2 AgeVehic 3 All Cens. cancel BM Formule AgeVehic

24 Results All Censoring Cancellation DurVie Mean of DurVie (in years) on January 1st, 1996 All Cens. cancel BM BM All Cens. cancel Formule Formule AgeVehic 1 AgeVehic 2 AgeVehic 3 All Cens. cancel BM Formule AgeVehic

25 Results coef exp(coef) se(coef) z p BM e+0000e+000 Formule e-003 Agevehic e-010 Rsquare= Likelihood ratio test= 174 on 3 df, p=0 Wald test = 174 on 3 df, p=0 Score (logrank) test = 179 on 3 df, p=0 Cox model

26 Results survival function BM1 BM2 BM time in years

27 Results survival functio on Agevehic1 Agevehic2 Agevehic3 Agevehic time in years

28 Results.0 survival function n Formule1 Formule2 Formule time in years

29 Results 0 BM1 BM2 BM3-6 log-log survival fu -4-2 nction time in years

30 Results log-log survival func ction Agevehic1 Agevehic2 Agevehic3 Agevehic time in years

31 Results nction log-log survival fu Formule1 Formule2 Formule time in years

32 Results Proportional hazard test: t g t 0 Test de: H = 0 avec gt ( ) t rho chisq p BM e-02 Formule e-04 Agevehic e-03 GLOBAL NA e-05

33 Results Schoenfeld residuals:

34 Results

35 Results

36 Results Additive Aalen Model Test for non-significant effects Supremum-test t of significance ifi p-value H_0: B(t)=0 (Intercept) Agevehic Formule BM Test for time invariant effects Kolmogorov-Smirnov test p-value H_0: B(t)=b t (Intercept) Agevehic Formule BM Cramer von Mises test p-value H_0: B(t)=b t (Intercept) Agevehic Formule BM

37 Results

38 Results

39 Results

40 Results

41 4- Conclusion and percpectives - Cox models with time-change covariates are not easy to understand or visualize. - Aelen model for failure time analysis allows the inclusion of time-dependent covariates as well as the variation of covariate effects over time. - Comparison with other models («duplication» models ) - Tests on another large dataset with new time dependant covariates..

42 Some References 1. Aalen, O.O. (1989). A linear regression model for the analysis of life times, Statistics in Medicine 8, Cox D.R. (1972). Regression models and life tables, J.R.Statist.Soc. B34, Grambsch P. and Therneau T.M. (1994). Proportional hazards tests and diagnostics based on weighted residuals. 3. Therneau T.M. and Grambsch P. (1990). Martingale-based residuals for survival models. Biometrika. 77, 1, pp

Applied Statistics. J. Blanchet and J. Wadsworth. Institute of Mathematics, Analysis, and Applications EPF Lausanne

Applied Statistics. J. Blanchet and J. Wadsworth. Institute of Mathematics, Analysis, and Applications EPF Lausanne Applied Statistics J. Blanchet and J. Wadsworth Institute of Mathematics, Analysis, and Applications EPF Lausanne An MSc Course for Applied Mathematicians, Fall 2012 Outline 1 Model Comparison 2 Model

More information

Statistics in Retail Finance. Chapter 6: Behavioural models

Statistics in Retail Finance. Chapter 6: Behavioural models Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics:- Behavioural

More information

SUMAN DUVVURU STAT 567 PROJECT REPORT

SUMAN DUVVURU STAT 567 PROJECT REPORT SUMAN DUVVURU STAT 567 PROJECT REPORT SURVIVAL ANALYSIS OF HEROIN ADDICTS Background and introduction: Current illicit drug use among teens is continuing to increase in many countries around the world.

More information

Checking proportionality for Cox s regression model

Checking proportionality for Cox s regression model Checking proportionality for Cox s regression model by Hui Hong Zhang Thesis for the degree of Master of Science (Master i Modellering og dataanalyse) Department of Mathematics Faculty of Mathematics and

More information

Modeling the Claim Duration of Income Protection Insurance Policyholders Using Parametric Mixture Models

Modeling the Claim Duration of Income Protection Insurance Policyholders Using Parametric Mixture Models Modeling the Claim Duration of Income Protection Insurance Policyholders Using Parametric Mixture Models Abstract This paper considers the modeling of claim durations for existing claimants under income

More information

Introduction to Event History Analysis DUSTIN BROWN POPULATION RESEARCH CENTER

Introduction to Event History Analysis DUSTIN BROWN POPULATION RESEARCH CENTER Introduction to Event History Analysis DUSTIN BROWN POPULATION RESEARCH CENTER Objectives Introduce event history analysis Describe some common survival (hazard) distributions Introduce some useful Stata

More information

Non-Parametric Estimation in Survival Models

Non-Parametric Estimation in Survival Models Non-Parametric Estimation in Survival Models Germán Rodríguez grodri@princeton.edu Spring, 2001; revised Spring 2005 We now discuss the analysis of survival data without parametric assumptions about the

More information

Introduction. Survival Analysis. Censoring. Plan of Talk

Introduction. Survival Analysis. Censoring. Plan of Talk Survival Analysis Mark Lunt Arthritis Research UK Centre for Excellence in Epidemiology University of Manchester 01/12/2015 Survival Analysis is concerned with the length of time before an event occurs.

More information

Survival Analysis Using SPSS. By Hui Bian Office for Faculty Excellence

Survival Analysis Using SPSS. By Hui Bian Office for Faculty Excellence Survival Analysis Using SPSS By Hui Bian Office for Faculty Excellence Survival analysis What is survival analysis Event history analysis Time series analysis When use survival analysis Research interest

More information

Tests for Two Survival Curves Using Cox s Proportional Hazards Model

Tests for Two Survival Curves Using Cox s Proportional Hazards Model Chapter 730 Tests for Two Survival Curves Using Cox s Proportional Hazards Model Introduction A clinical trial is often employed to test the equality of survival distributions of two treatment groups.

More information

200609 - ATV - Lifetime Data Analysis

200609 - ATV - Lifetime Data Analysis Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2015 200 - FME - School of Mathematics and Statistics 715 - EIO - Department of Statistics and Operations Research 1004 - UB - (ENG)Universitat

More information

Regression Modeling Strategies

Regression Modeling Strategies Frank E. Harrell, Jr. Regression Modeling Strategies With Applications to Linear Models, Logistic Regression, and Survival Analysis With 141 Figures Springer Contents Preface Typographical Conventions

More information

Survival Analysis of Left Truncated Income Protection Insurance Data. [March 29, 2012]

Survival Analysis of Left Truncated Income Protection Insurance Data. [March 29, 2012] Survival Analysis of Left Truncated Income Protection Insurance Data [March 29, 2012] 1 Qing Liu 2 David Pitt 3 Yan Wang 4 Xueyuan Wu Abstract One of the main characteristics of Income Protection Insurance

More information

SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

More information

Comparison of resampling method applied to censored data

Comparison of resampling method applied to censored data International Journal of Advanced Statistics and Probability, 2 (2) (2014) 48-55 c Science Publishing Corporation www.sciencepubco.com/index.php/ijasp doi: 10.14419/ijasp.v2i2.2291 Research Paper Comparison

More information

Survival Analysis, Software

Survival Analysis, Software Survival Analysis, Software As used here, survival analysis refers to the analysis of data where the response variable is the time until the occurrence of some event (e.g. death), where some of the observations

More information

Data Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression

Data Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction

More information

Developing Business Failure Prediction Models Using SAS Software Oki Kim, Statistical Analytics

Developing Business Failure Prediction Models Using SAS Software Oki Kim, Statistical Analytics Paper SD-004 Developing Business Failure Prediction Models Using SAS Software Oki Kim, Statistical Analytics ABSTRACT The credit crisis of 2008 has changed the climate in the investment and finance industry.

More information

Survival Analysis of Dental Implants. Abstracts

Survival Analysis of Dental Implants. Abstracts Survival Analysis of Dental Implants Andrew Kai-Ming Kwan 1,4, Dr. Fu Lee Wang 2, and Dr. Tak-Kun Chow 3 1 Census and Statistics Department, Hong Kong, China 2 Caritas Institute of Higher Education, Hong

More information

**BEGINNING OF EXAMINATION** The annual number of claims for an insured has probability function: , 0 < q < 1.

**BEGINNING OF EXAMINATION** The annual number of claims for an insured has probability function: , 0 < q < 1. **BEGINNING OF EXAMINATION** 1. You are given: (i) The annual number of claims for an insured has probability function: 3 p x q q x x ( ) = ( 1 ) 3 x, x = 0,1,, 3 (ii) The prior density is π ( q) = q,

More information

7.1 The Hazard and Survival Functions

7.1 The Hazard and Survival Functions Chapter 7 Survival Models Our final chapter concerns models for the analysis of data which have three main characteristics: (1) the dependent variable or response is the waiting time until the occurrence

More information

Nominal and ordinal logistic regression

Nominal and ordinal logistic regression Nominal and ordinal logistic regression April 26 Nominal and ordinal logistic regression Our goal for today is to briefly go over ways to extend the logistic regression model to the case where the outcome

More information

Distance to Event vs. Propensity of Event A Survival Analysis vs. Logistic Regression Approach

Distance to Event vs. Propensity of Event A Survival Analysis vs. Logistic Regression Approach Distance to Event vs. Propensity of Event A Survival Analysis vs. Logistic Regression Approach Abhijit Kanjilal Fractal Analytics Ltd. Abstract: In the analytics industry today, logistic regression is

More information

Statistical Analysis of Life Insurance Policy Termination and Survivorship

Statistical Analysis of Life Insurance Policy Termination and Survivorship Statistical Analysis of Life Insurance Policy Termination and Survivorship Emiliano A. Valdez, PhD, FSA Michigan State University joint work with J. Vadiveloo and U. Dias Session ES82 (Statistics in Actuarial

More information

Parametric Models. dh(t) dt > 0 (1)

Parametric Models. dh(t) dt > 0 (1) Parametric Models: The Intuition Parametric Models As we saw early, a central component of duration analysis is the hazard rate. The hazard rate is the probability of experiencing an event at time t i

More information

Interpretation of Somers D under four simple models

Interpretation of Somers D under four simple models Interpretation of Somers D under four simple models Roger B. Newson 03 September, 04 Introduction Somers D is an ordinal measure of association introduced by Somers (96)[9]. It can be defined in terms

More information

Lecture 15 Introduction to Survival Analysis

Lecture 15 Introduction to Survival Analysis Lecture 15 Introduction to Survival Analysis BIOST 515 February 26, 2004 BIOST 515, Lecture 15 Background In logistic regression, we were interested in studying how risk factors were associated with presence

More information

BayesX - Software for Bayesian Inference in Structured Additive Regression

BayesX - Software for Bayesian Inference in Structured Additive Regression BayesX - Software for Bayesian Inference in Structured Additive Regression Thomas Kneib Faculty of Mathematics and Economics, University of Ulm Department of Statistics, Ludwig-Maximilians-University Munich

More information

Tips for surviving the analysis of survival data. Philip Twumasi-Ankrah, PhD

Tips for surviving the analysis of survival data. Philip Twumasi-Ankrah, PhD Tips for surviving the analysis of survival data Philip Twumasi-Ankrah, PhD Big picture In medical research and many other areas of research, we often confront continuous, ordinal or dichotomous outcomes

More information

Structural Equation Models for Comparing Dependent Means and Proportions. Jason T. Newsom

Structural Equation Models for Comparing Dependent Means and Proportions. Jason T. Newsom Structural Equation Models for Comparing Dependent Means and Proportions Jason T. Newsom How to Do a Paired t-test with Structural Equation Modeling Jason T. Newsom Overview Rationale Structural equation

More information

An Application of Weibull Analysis to Determine Failure Rates in Automotive Components

An Application of Weibull Analysis to Determine Failure Rates in Automotive Components An Application of Weibull Analysis to Determine Failure Rates in Automotive Components Jingshu Wu, PhD, PE, Stephen McHenry, Jeffrey Quandt National Highway Traffic Safety Administration (NHTSA) U.S. Department

More information

SPSS TRAINING SESSION 3 ADVANCED TOPICS (PASW STATISTICS 17.0) Sun Li Centre for Academic Computing lsun@smu.edu.sg

SPSS TRAINING SESSION 3 ADVANCED TOPICS (PASW STATISTICS 17.0) Sun Li Centre for Academic Computing lsun@smu.edu.sg SPSS TRAINING SESSION 3 ADVANCED TOPICS (PASW STATISTICS 17.0) Sun Li Centre for Academic Computing lsun@smu.edu.sg IN SPSS SESSION 2, WE HAVE LEARNT: Elementary Data Analysis Group Comparison & One-way

More information

Exam C, Fall 2006 PRELIMINARY ANSWER KEY

Exam C, Fall 2006 PRELIMINARY ANSWER KEY Exam C, Fall 2006 PRELIMINARY ANSWER KEY Question # Answer Question # Answer 1 E 19 B 2 D 20 D 3 B 21 A 4 C 22 A 5 A 23 E 6 D 24 E 7 B 25 D 8 C 26 A 9 E 27 C 10 D 28 C 11 E 29 C 12 B 30 B 13 C 31 C 14

More information

Gamma Distribution Fitting

Gamma Distribution Fitting Chapter 552 Gamma Distribution Fitting Introduction This module fits the gamma probability distributions to a complete or censored set of individual or grouped data values. It outputs various statistics

More information

Cross-buying behaviour and customer loyalty in the insurance sector

Cross-buying behaviour and customer loyalty in the insurance sector 09 77 Cross-buying behaviour and customer loyalty in the insurance sector Guillén, M., Perch-Nielsen, J.; Pérez-Marín, A.M. (2009). Cross-buying behaviour and customer loyalty in the insurance sector.

More information

Modelling spousal mortality dependence: evidence of heterogeneities and implications

Modelling spousal mortality dependence: evidence of heterogeneities and implications 1/23 Modelling spousal mortality dependence: evidence of heterogeneities and implications Yang Lu Scor and Aix-Marseille School of Economics Lyon, September 2015 2/23 INTRODUCTION 3/23 Motivation It has

More information

Premaster Statistics Tutorial 4 Full solutions

Premaster Statistics Tutorial 4 Full solutions Premaster Statistics Tutorial 4 Full solutions Regression analysis Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression = 125,000 + 150. b. What is the prediction for

More information

Distribution (Weibull) Fitting

Distribution (Weibull) Fitting Chapter 550 Distribution (Weibull) Fitting Introduction This procedure estimates the parameters of the exponential, extreme value, logistic, log-logistic, lognormal, normal, and Weibull probability distributions

More information

Reliability Prediction for Mechatronic Drive Systems

Reliability Prediction for Mechatronic Drive Systems Reliability Prediction for Mechatronic Drive Systems Dipl.-Ing. Sebastian Bobrowski, Prof. Dr.-Ing. Wolfgang Schinköthe, University of Stuttgart, Institute of Design and Production in Precision Engineering

More information

5 Modeling Survival Data with Parametric Regression

5 Modeling Survival Data with Parametric Regression 5 Modeling Survival Data with Parametric Regression Models 5. The Accelerated Failure Time Model Before talking about parametric regression models for survival data, let us introduce the accelerated failure

More information

3. Regression & Exponential Smoothing

3. Regression & Exponential Smoothing 3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a

More information

e = random error, assumed to be normally distributed with mean 0 and standard deviation σ

e = random error, assumed to be normally distributed with mean 0 and standard deviation σ 1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.

More information

Statistics Graduate Courses

Statistics Graduate Courses Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.

More information

5. Linear Regression

5. Linear Regression 5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4

More information

Introduction to Survival Analysis

Introduction to Survival Analysis John Fox Lecture Notes Introduction to Survival Analysis Copyright 2014 by John Fox Introduction to Survival Analysis 1 1. Introduction I Survival analysis encompasses a wide variety of methods for analyzing

More information

Parametric survival models

Parametric survival models Parametric survival models ST3242: Introduction to Survival Analysis Alex Cook October 2008 ST3242 : Parametric survival models 1/17 Last time in survival analysis Reintroduced parametric models Distinguished

More information

The Cox Proportional Hazards Model

The Cox Proportional Hazards Model The Cox Proportional Hazards Model Mario Chen, PhD Advanced Biostatistics and RCT Workshop Office of AIDS Research, NIH ICSSC, FHI Goa, India, September 2009 1 The Model h i (t)=h 0 (t)exp(z i ), Z i =

More information

Multiple Linear Regression

Multiple Linear Regression Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is

More information

Didacticiel - Études de cas

Didacticiel - Études de cas 1 Topic Regression analysis with LazStats (OpenStat). LazStat 1 is a statistical software which is developed by Bill Miller, the father of OpenStat, a wellknow tool by statisticians since many years. These

More information

11. Analysis of Case-control Studies Logistic Regression

11. Analysis of Case-control Studies Logistic Regression Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:

More information

Regression. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Regression. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. Class: Date: Regression Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Given the least squares regression line y8 = 5 2x: a. the relationship between

More information

Yiming Peng, Department of Statistics. February 12, 2013

Yiming Peng, Department of Statistics. February 12, 2013 Regression Analysis Using JMP Yiming Peng, Department of Statistics February 12, 2013 2 Presentation and Data http://www.lisa.stat.vt.edu Short Courses Regression Analysis Using JMP Download Data to Desktop

More information

V. Kumar Andrew Petersen Instructor s Presentation Slides

V. Kumar Andrew Petersen Instructor s Presentation Slides V. Kumar Andrew Petersen Instructor s Presentation Slides Chapter Six Customer Churn 2 Introduction The most effective way to manage customer churn is to understand the causes or determinants of customer

More information

Wes, Delaram, and Emily MA751. Exercise 4.5. 1 p(x; β) = [1 p(xi ; β)] = 1 p(x. y i [βx i ] log [1 + exp {βx i }].

Wes, Delaram, and Emily MA751. Exercise 4.5. 1 p(x; β) = [1 p(xi ; β)] = 1 p(x. y i [βx i ] log [1 + exp {βx i }]. Wes, Delaram, and Emily MA75 Exercise 4.5 Consider a two-class logistic regression problem with x R. Characterize the maximum-likelihood estimates of the slope and intercept parameter if the sample for

More information

Basic Statistics and Data Analysis for Health Researchers from Foreign Countries

Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association

More information

Ordinal Regression. Chapter

Ordinal Regression. Chapter Ordinal Regression Chapter 4 Many variables of interest are ordinal. That is, you can rank the values, but the real distance between categories is unknown. Diseases are graded on scales from least severe

More information

Nonlinear Regression Functions. SW Ch 8 1/54/

Nonlinear Regression Functions. SW Ch 8 1/54/ Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General

More information

Basic Statistical and Modeling Procedures Using SAS

Basic Statistical and Modeling Procedures Using SAS Basic Statistical and Modeling Procedures Using SAS One-Sample Tests The statistical procedures illustrated in this handout use two datasets. The first, Pulse, has information collected in a classroom

More information

Parametric Survival Models

Parametric Survival Models Parametric Survival Models Germán Rodríguez grodri@princeton.edu Spring, 2001; revised Spring 2005, Summer 2010 We consider briefly the analysis of survival data when one is willing to assume a parametric

More information

Developing Risk Adjustment Techniques Using the SAS@ System for Assessing Health Care Quality in the lmsystem@

Developing Risk Adjustment Techniques Using the SAS@ System for Assessing Health Care Quality in the lmsystem@ Developing Risk Adjustment Techniques Using the SAS@ System for Assessing Health Care Quality in the lmsystem@ Yanchun Xu, Andrius Kubilius Joint Commission on Accreditation of Healthcare Organizations,

More information

Predicting Customer Default Times using Survival Analysis Methods in SAS

Predicting Customer Default Times using Survival Analysis Methods in SAS Predicting Customer Default Times using Survival Analysis Methods in SAS Bart Baesens Bart.Baesens@econ.kuleuven.ac.be Overview The credit scoring survival analysis problem Statistical methods for Survival

More information

Survival Analysis Approaches and New Developments using SAS. Jane Lu, AstraZeneca Pharmaceuticals, Wilmington, DE David Shen, Independent Consultant

Survival Analysis Approaches and New Developments using SAS. Jane Lu, AstraZeneca Pharmaceuticals, Wilmington, DE David Shen, Independent Consultant PharmaSUG 2014 - Paper PO02 Survival Analysis Approaches and New Developments using SAS Jane Lu, AstraZeneca Pharmaceuticals, Wilmington, DE David Shen, Independent Consultant ABSTRACT A common feature

More information

CS 688 Pattern Recognition Lecture 4. Linear Models for Classification

CS 688 Pattern Recognition Lecture 4. Linear Models for Classification CS 688 Pattern Recognition Lecture 4 Linear Models for Classification Probabilistic generative models Probabilistic discriminative models 1 Generative Approach ( x ) p C k p( C k ) Ck p ( ) ( x Ck ) p(

More information

Multivariable survival analysis S10. Michael Hauptmann Netherlands Cancer Institute Amsterdam, The Netherlands

Multivariable survival analysis S10. Michael Hauptmann Netherlands Cancer Institute Amsterdam, The Netherlands Multivariable survival analysis S10 Michael Hauptmann Netherlands Cancer Institute Amsterdam, The Netherlands m.hauptmann@nki.nl 1 Confounding A variable correlated with the variable of interest and with

More information

Regression Analysis: A Complete Example

Regression Analysis: A Complete Example Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

More information

EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION

EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION EDUCATION AND VOCABULARY 5-10 hours of input weekly is enough to pick up a new language (Schiff & Myers, 1988). Dutch children spend 5.5 hours/day

More information

Adequacy of Biomath. Models. Empirical Modeling Tools. Bayesian Modeling. Model Uncertainty / Selection

Adequacy of Biomath. Models. Empirical Modeling Tools. Bayesian Modeling. Model Uncertainty / Selection Directions in Statistical Methodology for Multivariable Predictive Modeling Frank E Harrell Jr University of Virginia Seattle WA 19May98 Overview of Modeling Process Model selection Regression shape Diagnostics

More information

Using An Ordered Logistic Regression Model with SAS Vartanian: SW 541

Using An Ordered Logistic Regression Model with SAS Vartanian: SW 541 Using An Ordered Logistic Regression Model with SAS Vartanian: SW 541 libname in1 >c:\=; Data first; Set in1.extract; A=1; PROC LOGIST OUTEST=DD MAXITER=100 ORDER=DATA; OUTPUT OUT=CC XBETA=XB P=PROB; MODEL

More information

Examining a Fitted Logistic Model

Examining a Fitted Logistic Model STAT 536 Lecture 16 1 Examining a Fitted Logistic Model Deviance Test for Lack of Fit The data below describes the male birth fraction male births/total births over the years 1931 to 1990. A simple logistic

More information

ESTIMATING THE DISTRIBUTION OF DEMAND USING BOUNDED SALES DATA

ESTIMATING THE DISTRIBUTION OF DEMAND USING BOUNDED SALES DATA ESTIMATING THE DISTRIBUTION OF DEMAND USING BOUNDED SALES DATA Michael R. Middleton, McLaren School of Business, University of San Francisco 0 Fulton Street, San Francisco, CA -00 -- middleton@usfca.edu

More information

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,

More information

Models Where the Fate of Every Individual is Known

Models Where the Fate of Every Individual is Known Models Where the Fate of Every Individual is Known This class of models is important because they provide a theory for estimation of survival probability and other parameters from radio-tagged animals.

More information

Comparison of sales forecasting models for an innovative agro-industrial product: Bass model versus logistic function

Comparison of sales forecasting models for an innovative agro-industrial product: Bass model versus logistic function The Empirical Econometrics and Quantitative Economics Letters ISSN 2286 7147 EEQEL all rights reserved Volume 1, Number 4 (December 2012), pp. 89 106. Comparison of sales forecasting models for an innovative

More information

A LONGITUDINAL AND SURVIVAL MODEL WITH HEALTH CARE USAGE FOR INSURED ELDERLY. Workshop

A LONGITUDINAL AND SURVIVAL MODEL WITH HEALTH CARE USAGE FOR INSURED ELDERLY. Workshop A LONGITUDINAL AND SURVIVAL MODEL WITH HEALTH CARE USAGE FOR INSURED ELDERLY Ramon Alemany Montserrat Guillén Xavier Piulachs Lozada Riskcenter - IREA Universitat de Barcelona http://www.ub.edu/riskcenter

More information

Chapter 4: Statistical Hypothesis Testing

Chapter 4: Statistical Hypothesis Testing Chapter 4: Statistical Hypothesis Testing Christophe Hurlin November 20, 2015 Christophe Hurlin () Advanced Econometrics - Master ESA November 20, 2015 1 / 225 Section 1 Introduction Christophe Hurlin

More information

Survival Distributions, Hazard Functions, Cumulative Hazards

Survival Distributions, Hazard Functions, Cumulative Hazards Week 1 Survival Distributions, Hazard Functions, Cumulative Hazards 1.1 Definitions: The goals of this unit are to introduce notation, discuss ways of probabilistically describing the distribution of a

More information

Lecture 2 ESTIMATING THE SURVIVAL FUNCTION. One-sample nonparametric methods

Lecture 2 ESTIMATING THE SURVIVAL FUNCTION. One-sample nonparametric methods Lecture 2 ESTIMATING THE SURVIVAL FUNCTION One-sample nonparametric methods There are commonly three methods for estimating a survivorship function S(t) = P (T > t) without resorting to parametric models:

More information

Least Squares Estimation

Least Squares Estimation Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

More information

Example 1: Calculate and compare RiskMetrics TM and Historical Standard Deviation Compare the weights of the volatility parameter using,, and.

Example 1: Calculate and compare RiskMetrics TM and Historical Standard Deviation Compare the weights of the volatility parameter using,, and. 3.6 Compare and contrast different parametric and non-parametric approaches for estimating conditional volatility. 3.7 Calculate conditional volatility using parametric and non-parametric approaches. Parametric

More information

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( ) Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

More information

Package depend.truncation

Package depend.truncation Type Package Package depend.truncation May 28, 2015 Title Statistical Inference for Parametric and Semiparametric Models Based on Dependently Truncated Data Version 2.4 Date 2015-05-28 Author Takeshi Emura

More information

Lecture 6: Poisson regression

Lecture 6: Poisson regression Lecture 6: Poisson regression Claudia Czado TU München c (Claudia Czado, TU Munich) ZFS/IMS Göttingen 2004 0 Overview Introduction EDA for Poisson regression Estimation and testing in Poisson regression

More information

Poisson Models for Count Data

Poisson Models for Count Data Chapter 4 Poisson Models for Count Data In this chapter we study log-linear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the

More information

Stat 5303 (Oehlert): Tukey One Degree of Freedom 1

Stat 5303 (Oehlert): Tukey One Degree of Freedom 1 Stat 5303 (Oehlert): Tukey One Degree of Freedom 1 > catch

More information

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not. Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

Is the Basis of the Stock Index Futures Markets Nonlinear?

Is the Basis of the Stock Index Futures Markets Nonlinear? University of Wollongong Research Online Applied Statistics Education and Research Collaboration (ASEARC) - Conference Papers Faculty of Engineering and Information Sciences 2011 Is the Basis of the Stock

More information

Exam Introduction Mathematical Finance and Insurance

Exam Introduction Mathematical Finance and Insurance Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closed-book exam. The exam does not use scrap cards. Simple calculators are allowed. The questions

More information

Lecture 14: GLM Estimation and Logistic Regression

Lecture 14: GLM Estimation and Logistic Regression Lecture 14: GLM Estimation and Logistic Regression Dipankar Bandyopadhyay, Ph.D. BMTRY 711: Analysis of Categorical Data Spring 2011 Division of Biostatistics and Epidemiology Medical University of South

More information

Distressed Debt Prices and Recovery Rate Estimation

Distressed Debt Prices and Recovery Rate Estimation Distressed Debt Prices and Recovery Rate Estimation Robert Jarrow Joint Work with Xin Guo and Haizhi Lin May 2008 Introduction Recent market events highlight the importance of understanding credit risk.

More information

Students' Opinion about Universities: The Faculty of Economics and Political Science (Case Study)

Students' Opinion about Universities: The Faculty of Economics and Political Science (Case Study) Cairo University Faculty of Economics and Political Science Statistics Department English Section Students' Opinion about Universities: The Faculty of Economics and Political Science (Case Study) Prepared

More information

5. Ordinal regression: cumulative categories proportional odds. 6. Ordinal regression: comparison to single reference generalized logits

5. Ordinal regression: cumulative categories proportional odds. 6. Ordinal regression: comparison to single reference generalized logits Lecture 23 1. Logistic regression with binary response 2. Proc Logistic and its surprises 3. quadratic model 4. Hosmer-Lemeshow test for lack of fit 5. Ordinal regression: cumulative categories proportional

More information

HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009

HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 1. Introduction 2. A General Formulation 3. Truncated Normal Hurdle Model 4. Lognormal

More information

Threshold Autoregressive Models in Finance: A Comparative Approach

Threshold Autoregressive Models in Finance: A Comparative Approach University of Wollongong Research Online Applied Statistics Education and Research Collaboration (ASEARC) - Conference Papers Faculty of Informatics 2011 Threshold Autoregressive Models in Finance: A Comparative

More information

Simple Methods and Procedures Used in Forecasting

Simple Methods and Procedures Used in Forecasting Simple Methods and Procedures Used in Forecasting The project prepared by : Sven Gingelmaier Michael Richter Under direction of the Maria Jadamus-Hacura What Is Forecasting? Prediction of future events

More information

Applying Survival Analysis Techniques to Loan Terminations for HUD s Reverse Mortgage Insurance Program - HECM

Applying Survival Analysis Techniques to Loan Terminations for HUD s Reverse Mortgage Insurance Program - HECM Applying Survival Analysis Techniques to Loan Terminations for HUD s Reverse Mortgage Insurance Program - HECM Ming H. Chow, Edward J. Szymanoski, Theresa R. DiVenti 1 I. Introduction "Survival Analysis"

More information

Lecture 4 PARAMETRIC SURVIVAL MODELS

Lecture 4 PARAMETRIC SURVIVAL MODELS Lecture 4 PARAMETRIC SURVIVAL MODELS Some Parametric Survival Distributions (defined on t 0): The Exponential distribution (1 parameter) f(t) = λe λt (λ > 0) S(t) = t = e λt f(u)du λ(t) = f(t) S(t) = λ

More information

Personalized Predictive Medicine and Genomic Clinical Trials

Personalized Predictive Medicine and Genomic Clinical Trials Personalized Predictive Medicine and Genomic Clinical Trials Richard Simon, D.Sc. Chief, Biometric Research Branch National Cancer Institute http://brb.nci.nih.gov brb.nci.nih.gov Powerpoint presentations

More information

E(y i ) = x T i β. yield of the refined product as a percentage of crude specific gravity vapour pressure ASTM 10% point ASTM end point in degrees F

E(y i ) = x T i β. yield of the refined product as a percentage of crude specific gravity vapour pressure ASTM 10% point ASTM end point in degrees F Random and Mixed Effects Models (Ch. 10) Random effects models are very useful when the observations are sampled in a highly structured way. The basic idea is that the error associated with any linear,

More information

Linda Staub & Alexandros Gekenidis

Linda Staub & Alexandros Gekenidis Seminar in Statistics: Survival Analysis Chapter 2 Kaplan-Meier Survival Curves and the Log- Rank Test Linda Staub & Alexandros Gekenidis March 7th, 2011 1 Review Outcome variable of interest: time until

More information